The Law of Truly Large Numbers and Ancestors?

i'm anthropology student studying inter-relatedness of humans. i'm using statistics, law of large numbers.
according law of large numbers... "that sample size large enough, outrageous thing happen.
means "unlikely event" has probability of 63.2% of happening if 1000 chances given, or on 99.9% 10,000 chances. in other words, highly unlikely event, given enough tries, more unlikely not occur."

law of large numbers, not own idea. have below computer scientist:

if there random intermixing, each have ~1 million ancestors living in 1500 ad, out of world population of ~500 million.
fractional overlap between 2 people 1/500th.

probability 2 people share @ least 1 common ancestor 100%. basically, choosing random number between 1 , 500 million times , you're asking whether ever choose number 500. in million trials, expect happen 2000 times. happens @ least once guaranteed.

if rid of random intermixing, fractional overlap drop less 1/500th. suspect probability of @ least 1 overlap remain high.
if population in 1500 500 million, , 6 billion today (12x larger).
if average generation length 30 years, there 17 generations in 500 years.
average number of surviving children per mother exp((log 12)/17) = 1.157
since child has 2 parent, average number of surviving children per person 2 * 1.157 = 2.315
average growth rate per generation descendants of person in 1500.
2.315^17 = 1.575 million. average person in 1500 has 1.5 million offspring alive today. sampling whole world, probability random person 1500 ancestor of random person in 2000 1.5 million / 6 billion = 0.025%.
if considering people in region europe, 1.4 million / 700 million = 0.2%
here theory:

lets 500 years ago, due probabilistic law, alive today had @ least 1 ancestor say, china, or whatever remote region had @ least contact other places.
theory true given enough time?

i'm anthropology student studying inter-relatedness of humans. i'm using statistics, law of large numbers. according law of large numbers... "that sample size large enough, outrageous thing happen. means this...

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